Explanation
Let W be a vector space, with an inner product . For every linear function there exists a unique vector v in W such that for all w in W. The map given by is an isomorphism. This holds for all vector spaces, and can be used to explain the Hodge dual.
Let V be an n-dimensional vector space with basis . For 0 ≤ k≤ n, consider the exterior power spaces and . For each and, we have . There is, up to a scalar, only one n-vector, namely . In other words, must be a scalar multiple of for all and .
Consider a fixed . There exists a unique linear function such that for all . This is the scalar multiple mentioned in the previous paragraph. If denotes the inner product on (n–k)-vectors, then there exists a unique (n–k)-vector, say, such that for all . This (n–k)-vector is the Hodge dual of λ, and is the image of the under the canonical isomorphism between and . Thus, .
Read more about this topic: Hodge Dual
Famous quotes containing the word explanation:
“The explanation of the propensity of the English people to portrait painting is to be found in their relish for a Fact. Let a man do the grandest things, fight the greatest battles, or be distinguished by the most brilliant personal heroism, yet the English people would prefer his portrait to a painting of the great deed. The likeness they can judge of; his existence is a Fact. But the truth of the picture of his deeds they cannot judge of, for they have no imagination.”
—Benjamin Haydon (1786–1846)
“Are cans constitutionally iffy? Whenever, that is, we say that we can do something, or could do something, or could have done something, is there an if in the offing—suppressed, it may be, but due nevertheless to appear when we set out our sentence in full or when we give an explanation of its meaning?”
—J.L. (John Langshaw)
“There is a great deal of unmapped country within us which would have to be taken into account in an explanation of our gusts and storms.”
—George Eliot [Mary Ann (or Marian)